Existence Of Solutions For Several Classes Of Elliptic Partial Differential Equations

Partial differential equations are often based on practical problems in physics,biology engineering and other disciplines.PDEs are a practical subject,which has attracted wide attention from scholars at home and abroad and have achieved fruitful research results.Elliptic partial differential equation is an important branch in the field of partial differential equation.The existence and nonexistence of its solution is one of the main research contents in this field.In this thesis,we study the existence and multiplicity of nontrivial weak solutions of two kinds of elliptic partial differential equations.We mainly use the variational method to transform the problem into the problem of finding the critical point of energy functional corresponding to the equation,and then prove the existence of the critical point of energy functional with the help of Mountain Pass Theorem and Ekland variational principle.In Chapter 2,we study the multiplicity of solutions for the nonhomogeneous p-Kichhoff elliptic equation???where ??? is the complement of a smooth bounded domain D in???.h>0,M?s?=a+bs^k,a,b>0,k?0,k 30,h₁?x?,h₂?x?,h₃?x?are contiuous functions which may change sign on ???.The parameters p,q,r satisfy1<q<p?k+1?<r<p^*=Np/?N-p?.A new existence result for multiple solutions is obtained by the Mountain pass theorem and Ekeland's variational principle.In Chapter 3,we consider the quasilinear elliptic equation:???in???,where 1?p?N and ??? is a nonlinearity depending also on the gradient of the solution.The existence of a positive solution is stated through an itertive method based on Mountain Pass Techniques.In Chapter 4,we summarize the main work of this paper and look forward to the future research work.